optimal control of systems governed by pdes with random ...optimal control of systems governed by...

Optimal control of systems governed by PDEs withrandom parameter fields using quadratic approximations

Alen Alexanderian,1 Omar Ghattas,2 Noemi Petra3

Georg Stadler,4 Umberto Villa2

1Department of MathematicsNorth Carolina State University

2Institute for Computational Engineering and SciencesThe University of Texas at Austin

3School of Natural SciencesUniversity of California, Merced

3Courant Institute of Mathematical SciencesNew York University

Computational Methods for Control of Infinite-dimensional SystemsInstitute for Mathematics and its Applications

University of Minnesota, Minneapolis, MinnesotaMarch 14, 2016

Optimal control of systems governed by PDEs withuncertain parameter fields

PDE-constrained control objective

q = q(u(z,m))

where

A(u,m) = f(z)

q: control objective

A: forward operator

m: uncertain parameter field

z: control function

Control of injection wells in porousmedium flow (SPE10 permeability data)

Problem: given the uncertainty model for m, find z that “optimizes” q(u(z,m))

Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 2 / 28

Optimization under uncertainty (OUU)

H : parameter space, infinite-dimensional separable Hilbert space

q(z,m): control objective functional

m ∈H : uncertain model parameter field, z: control function

Optimization under uncertainty (OUU):

minz

Em{

q(z,m)

}+ β varm{q(z,m)}

Em{q(z,m)} =

∫H

q(z,m)µ(dm)

varm{q(z,m)} = Em{q(z,m)2} − Em{q(z,m)}2

Main challenges:

Integration over infinite/high-dimensional parameter spaceEvaluation of q requires PDE solves

Standard Monte Carlo approach (Sample Average Approximation) isprohibitive

Numerous (nmc) samples required, each requires PDE solveResulting PDE-constrained optimization problem has nmc PDE constraints






Risk-neutral optimization under uncertainty (OUU):

minz

Em{q(z,m)}

+ β varm{q(z,m)}

Em{q(z,m)} =

∫H

q(z,m)µ(dm)


Main challenges:









Risk-averse optimization under uncertainty (OUU):

minz

Em{q(z,m)}+ β varm{q(z,m)}

Em{q(z,m)} =

∫H

q(z,m)µ(dm)


Main challenges:





Some existing approaches for PDE-constrained OUU

Methods based on stochastic collocation, sparse/adaptive sampling, POD, ...Schulz & Schillings, Problem formulations and treatment of uncertainties in aerodynamic design, AIAAJ, 2009.Borzı & von Winckel, Multigrid methods and sparse-grid collocation techniques for parabolic optimalcontrol problems with random coefficients, SISC, 2009.Borzı, Schillings, & von Winckel, On the treatment of distributed uncertainties in PDE-constrainedoptimization, GAMM-Mitt. 2010.Borzı & von Winckel, A POD framework to determine robust controls in PDE optimization, Computingand Visualization in Science, 2011.Gunzburger & Ming, Optimal control of stochastic flow over a backward-facing step using reduced-ordermodeling, SISC 2011.Gunzburger, Lee, & Lee, Error estimates of stochastic optimal Neumann boundary control problems,SINUM, 2011.Kunoth & Schwab, Analytic Regularity and GPC Approximation for Control Problems Constrained byLinear Parametric Elliptic and Parabolic PDEs, SICON, 2013.Tiesler, Kirby, Xiu, & Preusser, Stochastic collocation for optimal control problems with stochastic PDEconstraints, SICON, 2012.Kouri, Heinkenschloss, Ridzal, & Van Bloemen Waanders, A trust-region algorithm with adaptivestochastic collocation for PDE optimization under uncertainty, SISC, 2013.Chen, Quarteroni, & Rozza, Stochastic optimal Robin boundary control problems ofadvection-dominated elliptic equations, SINUM, 2013.Kouri, A multilevel stochastic collocation algorithm for optimization of PDEs with uncertain coefficients,JUQ, 2014.Chen & Quarteroni, Weighted reduced basis method for stochastic optimal control problems with ellipticPDE constraint, JUQ, 2014.Chen, Quarteroni, & Rozza, Multilevel and weighted reduced basis method for stochastic optimalcontrol problems constrained by Stokes equations, Num. Math. 2015.


Control of injection wells in a porous medium flow

1

2

3

4

m = mean of log permeability field q = target pressure at production wells

state PDE: single phase flow in a porous medium

−∇ · (em∇u) =

nc∑i=1

zifi(x)

with Dirchlet lateral & Neumann top/bottom BCs

uncertain parameter: log permeability field m

control variables: zi, mass source at injection wells; fi, mollified Dirac deltas

control objective: q(z,m) := 12‖Qu(z,m)− q‖2, q: target pressure

dimensions: ns = nm = 3242, nc = 20, nq = 12


Porous medium with random permeability field

Distribution law of m:

µ = N (m, C) (Gaussian measure on Hilbert space H )

Take covariance operator as square of inverse of Poisson-like operator:

C = (−κ∆ + αI)−2 κ, α > 0

C is positive, self-adjoint, of trace-class; µ well-defined on H (Stuart ’10)κα ∝ correlation length; the larger α, the smaller the variance

Random draws for κ = 2× 10−2, α = 4


OUU with linearized parameter-to-objective map

Risk-averse optimal control problem (including cost of controls):

minz

Em{q(z,m)}+ β varm{q(z,m)}+ γ ‖z‖2

Linear approximation to parameter-to-objective map about m:

qlin(z,m) = q(z, m) + 〈gm(z, m),m− m〉

gm(z, ·) :=dq(z, ·)dm

is the gradient with respect to m

The moments of the linearized objective:

Em{qlin(z, ·)} = q(z, m),

varm{qlin(z, ·)} = 〈gm(z, m), C[gm(z, m)]〉

qlin(z, ·) ∼ N(q(z, m), 〈gm(z, m), C[gm(z, m)]〉

)Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 7 / 28

Risk-averse optimal control problem with linearizedparameter-to-objective map

State-and-adjoint-PDE constrained optimization problem (quartic in z):

minz∈ZJ (z) :=

1

2‖Qu− q‖2 +

β

2〈gm(m), C[gm(m)]〉+

γ

2‖z‖2

with gm(m) =em∇u · ∇p, where

−∇ · (em∇u) =

nc∑i=1

zifi state equation

−∇ · (em∇p) = −Q∗(Qu− q) adjoint equation

Lagrangian of the risk-averse optimal control problem with qlin:

L (z, u, p, u?, p?) =1

2‖Qu− q‖2 +

β

2

⟨em∇u · ∇p, C[em∇u · ∇p]

⟩+γ

2‖z‖2

+⟨em∇u,∇u?⟩− nc∑

i=1

zi〈fi, u?〉

+⟨em∇p,∇p?

⟩+ 〈Q∗(Qu− q), p?〉


Gradient computation for risk averse optimal control

“State problem” for risk-averse optimal control problem with qlin:

⟨em∇u,∇u

⟩=

nc∑i=1

zi〈fi, u〉⟨em∇p,∇p

⟩= −〈Q∗(Qu− q), p〉

for all test functions p and u

“Adjoint problem” for risk-averse optimal control problem with qlin:

⟨em∇p?,∇p

⟩= −β

⟨em∇u · ∇p, C[em∇u · ∇p]

⟩⟨em∇u?,∇u

⟩= −〈Q∗(Qu− q), u〉 − β

⟨em∇u · ∇p, C[em∇u · ∇p]

⟩− 〈Q∗Qp?, u〉

for all test functions p and u

Gradient:∂J∂zj

= γzj − 〈fj , u?〉, j = 1, . . . , nc

Cost of objective = 2 PDE solves; cost of gradient = 2 PDE solves


Risk-averse optimal control with linearized objective

−20 0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

dis

trib

uti

on

Θ(z0, ·)Θlin(z0, ·)Θquad(z0, ·)

initial (suboptimal) control z0 distrib. of exact & approx objectives at z0

0 2 4 6 8 10 12 14 160

1

2

3

4

5

dis

trib

uti

on

Θ(zoptlin , ·)

Θlin(zoptlin , ·)

Θquad(zoptlin , ·)

optimal control zoptlin based on qlin distrib. of exact & approx objectives at zopt

lin


Quadratic approximation to parameter-to-objective map

Quadratic approximation to the parameter-to-control-objective map:

qquad(z,m) = q(z, m) + 〈gm(z, m),m− m〉+1

2〈Hm(z, m)(m− m),m− m〉

gm: gradient of parameter-to-objective map

Hm: Hessian of parameter-to-objective map

Observations:

Quadratic approximation does not lead to a Gaussian control objective

However, can derive analytic formulas for the moments of qquad in theinfinite-dimensional Hilbert space setting


Analytic expressions for mean and variance with qquad

Mean:

Em{qquad(z, ·)} = q(z, m)+1

2tr[Hm(z, m)

]Variance:

varm{qquad(z, ·)} = 〈gm(z, m), C[gm(z, m)]〉+1

2tr[Hm(z, m)2

]where Hm = C1/2HmC1/2 is the covariance-preconditioned Hessian

Risk averse optimal control objective with qquad:

J(z) = q(z, m) +1

2tr[Hm(z, m)

]+β

2

{〈gm(z, m), C[gm(z, m)]〉+

1

2tr[Hm(z, m)2

]}


Randomized trace estimator

Randomized trace estimation:

tr(Hm) ≈ 1

ntr

ntr∑j=1

⟨Hmξj , ξj

⟩=

1

ntr

ntr∑j=1

〈Hmζj , ζj〉

tr(H2m) ≈ 1

ntr

ntr∑j=1

〈Hmζj , C[Hmζj ]〉

where ξj are random Gaussian fields and ζj = C1/2ξj

In computations, we use draws ζj ∼ N (0, C) =: ν

Straightforward to show:∫H

〈Hmζ, ζ〉 ν(dζ) = tr(Hm),

∫H

〈Hmζ, C[Hmζ]〉 ν(dζ) = tr(H2m)

Finite dimensional algorithm: H. Avron and S. Toledo, Randomizedalgorithms for estimating the trace of an implicit symmetric positivesemi-definite matrix, Journal of the ACM, 2011.


Risk-averse optimal control with quadraticized objective

minz∈Z

1

2‖Qu− q‖22+

1

2ntr

ntr∑j=1

〈ζj , ηj〉+β

2

{〈gm(m), C[gm(m)]〉+ 1

2ntr

ntr∑j=1

∥∥C1/2ηj∥∥2}

with

gm(m) =em∇u · ∇pηj = em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)︸︷︷︸

Hmζj

j ∈ {1, . . . , ntr}

where

−∇ · (em∇u) =∑Ni=1 zifi

−∇ · (em∇p) = −Q∗(Qu− q)−∇ · (em∇υj) = ∇ · (ζjem∇u)

−∇ · (em∇ρj) = −Q∗Qυj +∇ · (ζjem∇p)


Lagrangian for risk-averse optimal control with qquad

L (z, u, p,{υj}ntrj=1, {ρj}ntr

j=1, u?, p?, {υ?

j}ntrj=1, {ρ

?j}ntr

j=1)

=1

2‖Qu− q‖22

+1

2ntr

ntr∑j=1

⟨ζj ,[em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)

]⟩+β

2

⟨em∇u · ∇p, C[em∇u · ∇p]

⟩+

β

4ntr

ntr∑j=1

∥∥∥C1/2 [em(ζj∇u · ∇p+∇υj · ∇p+∇u · ∇ρj)]∥∥∥2

+⟨em∇u,∇u?⟩−∑N

i=1 zi〈fi, u?〉

+⟨em∇p,∇p?

⟩+ 〈Q∗(Qu− q), p?〉

+

ntr∑j=1

[⟨em∇υj ,∇υ?

j

⟩+⟨ζje

m∇u,∇υ?j

⟩]+

ntr∑j=1

[⟨em∇ρj ,∇ρ?

j

⟩+⟨Q∗Qυj , ρ

?j

⟩+⟨ζje

m∇p,∇ρ?j

⟩]Omar Ghattas (ICES, UT Austin) Optimal control under uncertainty March 14, 2016 15 / 28

Adjoint & gradient for risk-averse optimal control w/qquad

Adjoint problem for qquad approximation

−∇ · (em∇ρ?j ) = b(j)1 j ∈ {1, . . . , ntr}

− ∇ · (em∇υ?j ) +Q∗Qρ?j = b(j)2 j ∈ {1, . . . , ntr}

− ∇ · (em∇p?)−ntr∑j=1

∇ · (ζjem∇ρ?j ) = b3

−∇ · (em∇u?) +Q∗Qp? −ntr∑j=1

∇ · (ζjem∇υ?j ) = b4

Gradient for qquad approximation

∂L

∂zj= γzj − 〈fj , u?〉, j = 1, . . . , nc

Cost of objective = 2 + 2ntr PDE solves; cost of gradient = 2 + 2ntr PDE solves


Risk-averse optimal control with quadraticized objective

−20 0 20 40 60 80 100 120 1400

0.01

0.02

0.03

0.04

dis

trib

uti

on

Θ(z0, ·)Θlin(z0, ·)Θquad(z0, ·)

initial (suboptimal) control z0 distrib. of exact & approx objectives at z0

0 2 4 6 8 10 12 14 160

0.2

0.4

0.6

dis

trib

uti

on

Θ(zoptquad, ·)

Θquad(zoptquad, ·)

optimal control zoptquad based on qquad distrib. of exact & approx objectives at zopt

quad


Effect of number of trace estimator vectors on distributionof control objective evaluated at optimal controls

0 1 2 3 4 5 6 7 80

0.2

0.4

value of control objective

distribution

ntr = 5ntr = 20ntr = 40ntr = 60

Optimal controls zoptquad computed for each value of trace estimator using quadratic

approximation of control objective, qquad

Each curve based on 10,000 samples of distribution of q(zoptquad,m)

(control objective evaluated at optimal control)


Comparison of distribution of control objective for optimalcontrols based on linearized and quadraticized objective

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

value of control objective

dis

trib

uti

on

Θ(zoptquad, ·)

Θ(zoptlin , ·)

1000 5000 100001

2

3

4

5

sample size

sample

mean

1000 5000 10000

5

10

15

sample size

sample

variance

Comparison of the distributions of q(zoptlin ,m) with q(zopt

quad,m)

β = 1, γ = 10−5 and ntr = 40 trace estimation vectors

KDE results are based on 10,000 samples

Inserts show Monte Carlo sample convergence for mean and variance


Some concluding remarks

Optimal control of PDEs with infinite-dimensional parameters

Use of local approximations to parameter-to-objective map

Quadratic approximation is effective in capturing the distribution of thecontrol objective for the target problem

Computational complexity of objective & gradient evaluation is independentof problem dimension (nm = 3242)

objective + gradient cost = ∼125 state PDE solvesobjective/gradient cost for Monte Carlo/SAA: ∼10,000 state PDE solvesdifferences even more striking for nonlinear PDE state problems, since all butone PDE solves are linearized (with SAA all are nonlinear)

For further details, see:

A. Alexanderian, N. Petra, G. Stadler, and O. Ghattas, Mean-variancerisk-averse optimal control of systems governed by PDEs with randomcoefficient fields using second order approximations, submitted.arXiv:1602.07592

Research funded by DOE MMICCs (DE-FC02-13ER26128) and DARPA EQUiPS(W911NF-15-2-0121).


Extensions

Alternative risk measures

Higher moments

Higher order approximations of parameter-to-objective map, e.g., third orderTaylor expansion with proper tensor contraction of third order derivativetensor

Highly nonlinear/complex forward problems

Monte Carlo corrections to quadratic approximation


Reynolds averaged Navier-Stokes model for a turbulent jet

Governing equations: k-ε closure model

(∇u)u−∇ ·[(ν + νt)

(∇u +∇ut)− p I] = 0

∇ · u = 0

−∇ ·[(ν +

νtσk

)∇k]

+ u · ∇k − 1

2νttr

[(∇u +∇ut)2]+ ε = 0

−∇ ·[(ν +

νtσε

)∇ε]

+ u · ∇ε− 1

2Cε1

ε

kνttr

[(∇u +∇ut)2]+ Cε2

ε2

k= 0

where νt = Cµk2

ε

Quantity of Interest: Jet width at a given point x = x0

Q =

∫Γ0

u · e1dy

ucl(x0)

where Γ0 is the cross-section at x = x0 and ucl(x0) is the centerline velocity


RANS model for a turbulent jet

State DOF = 28,379; uncertain parameter DOF = 2626; O(100) damped Newtonsteps per nonlinear state solve (10 min per state solve)


Reynolds averaged Navier-Stokes inadequacy model

To model inadequacy of the k-ε closure model we introduce an uncertainprefactor in the turbulent viscosity of the form

νt = emCµk2

ε

Here m ∼ N (0, C) is a spatially correlated normal random field with acovariance operator of the form

C = (δ − γ∇ ·Θ∇)−2

where Θ is an s.p.d tensor that allows for anisotropy in the correlation length

Figure: Samples from m ∼ N (0, C)


RANS turbulent jet: Spectrum of the Hessian

Figure: Spectrum of the Hessian Hm (left) and of the prior preconditioned HessianHm = C1/2HmC1/2 (right)


RANS turbulent jet: Linear and quadratic approximation

To estimate the expected value of the q.o.i. Q we used a Monte Carloestimator QMC (500 samples, estimator variance 1.49× 10−5).

We then used the analytical expression presented before to compute theexpectation of the linearized Qlin and quadratic Qquad approximation of theq.o.i.

We finally considered an approximation Qquad of the Qquad that uses a lowrank approximation HLRm of the Hessian instead of the the true Hessian Hm.The main advantage is that now the trace of C1/2HLRm C1/2 can becomputed analytically and that sampling Qquad does not require to solve anyincremental forward/adjoint problem.

QMC E[Qlin] E[Qquad] E[Qquad]1.2044 1.2010 1.2125 1.2112


RANS turbulent jet: Variance reduction MC

The linear and quadratic approximations of Q are effective (and easy tocompute) surrogates for variance reduction in MC simulations

In fact, by linearity of expectation, for Q = {Qlin, Qquad, Qquad} and

Y = Q− Q we have

E[Q] = E[Q− Q] + E[Q] = E[Y ] + E[Q]

Since Var[Y ]� Var[Q], many fewer samples are required to achieve thesame variance as that of the Monte Carlo estimator

Q Ylin Yquad Yquad

Variance 7.4670E-03 1.3426E-04 3.7904E-05 3.2583E-05Number of Samples 7467 134 38 33

Table: Variance of the Monte Carlo estimators for Q and the correction Y . The secondrow indicates the number of samples necessary to obtain a mean square error of 10−6


Reynolds Averaged Navier-Stokes - Variance Reduction MC

n samples QMC Y MClin + E[Qlin] Y MC

quad + E[Qquad] Y MCquad + E[Qquad]

50 1.2157 1.2086 1.2108 1.2097100 1.2129 1.2079 1.2112 1.2101150 1.2107 1.2088 1.2105 1.2095200 1.2078 1.2092 1.2100 1.2090250 1.2063 1.2091 1.2101 1.2091300 1.2055 1.2087 1.2102 1.2092350 1.2036 1.2086 1.2103 1.2092400 1.2036 1.2083 1.2104 1.2093450 1.2030 1.2082 1.2103 1.2092500 1.2044 1.2084 1.2102 1.2092

n samples V ar[QMC ] V ar[Y MClin ] V ar[Y MC

quad] V ar[Y MCquad]

50 1.6866E-04 5.3357E-06 4.7332E-07 4.1531E-07100 7.4525E-05 1.6918E-06 2.9706E-07 2.5409E-07150 5.6089E-05 1.3427E-06 2.3033E-07 1.9537E-07200 4.2830E-05 1.0260E-06 2.3183E-07 2.0324E-07250 3.3459E-05 7.2285E-07 1.6875E-07 1.4784E-07300 2.6100E-05 5.3132E-07 1.3172E-07 1.1483E-07350 2.2158E-05 4.4029E-07 1.1049E-07 9.4659E-08400 1.8295E-05 3.4960E-07 8.9182E-08 7.6542E-08450 1.6088E-05 2.8972E-07 7.8000E-08 6.6891E-08500 1.4934E-05 2.6852E-07 7.5808E-08 6.5166E-08


optimal control of systems governed by pdes with random ...optimal control of systems governed by...

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